Basic concepts for Localization of deformation. Stress vs. displacement/velocity boundary conditions - unstable/stable processes

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1 Basic concepts for Localization of deformation Weakening vs. strengthening rheologies (P, T, porosity, fluids, grain size) positive vs. negative feedbacks Stress vs. displacement/velocity boundary conditions - unstable/stable processes Effects of healing for displacement/velocity boundary conditions - ratio of healing/loading timescales Effects of heterogeneities (can suppress localization) Effects of inherited structures: bimaterial interfaces, weak zones,... Effective behavior on different scales: single microcrack, macroscopic shear crack, cataclastic flow, fault zone, fault system, plate boundary,

2 Weakening vs. strengthening rheologies (P, T, porosity, fluids, grain size) positive vs. negative feedback mechanisms

3 Deformation band (Shear band) Deformation zone Goblin Valley, Utah

4 The Punchbowl fault: 44 km of slip (Chester and Chester, 8)

5 Stress vs. displacement/velocity boundary conditions - unstable/stable processes

6 Stability Constant stress loading lead to dynamic instability once a critical crack length L c has been archived (perhaps by creep). Constant stress Griffith experiments unstable a K τ R Constant stress In contrast, under constant displacement/velocity, failure is stable stable a Constant displacement loading Stability is NOT a material property but the response of the entire system (failure zone plus loading environment). Stability is determined by comparing rate of strength change in failure area with rate of loading reduction during failure (zero in a, finite in b). The latter is referred to as the stiffness of the system.

7 Effects of healing for displacement/velocity boundary conditions - ratio of healing/loading timescales

8 Coupled evolution of earthquakes and faults Distributed Damage (fault zones) H = km ν = 0.2 Evolving Elastic Upper Crust h = 20 km Loading by distributed steady mantle motion Viscoelastic Mantle (half space) Viscoelastic Lower Crust 00 km We fix all the large scale parameters (e.g., dimensions, background elastic properties, viscosity) using data associated with the San Andreas fault. The evolving results depend on the ratio of time scale for damage healing τ H to time scale for tectonic loading τ L

9 slow effective healing fast effective healing α Characteristic earthquakes Power-law statistics

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11 Effects of inherited structures: bimaterial interfaces, weak zones,...

12 Weertman (80): 2D analytical solution for steady state mode II slip pulse on a bimaterial interface governed by Coulomb friction. In-plane slip: δ(x,t)= u(x,y= 0 +,t) u(x,y= 0,t) Moving coordinate system: ξ = x ct Dislocation density: B(ξ) = dδ/dξ The shear and normal stress on the interface are μ ( c, Δβ ) B( ξ ') τ ( ξ ) = τ + dξ ' π ξ ξ ' σ ( ξ ) = σ μ *( c, Δβ ) B( ξ ) compliant stiff u(x,y = 0 +,t) u(x,y = 0,t) In a homogeneous solid μ* = 0; there is no coupling between slip and σ. For subsonic rupture on a bimaterial interface in the direction of motion of the compliant solid, μ*> 0 and σ drops dynamically (producing local dilation). In the opposite direction, μ*< 0 and σ increases dynamically (local compression). Adams (): The bimaterial effects increase with propagation distance!

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14 Wrinkle-like rupture on a bimaterial interface compliant stiff Andrews and Ben-Zion, ; Ben-Zion and Andrews, 8; Cochard and Rice, 2000; Ben-Zion, 200; Ben-Zion and Huang, 2002, Shi and Ben-Zion, 2006; Ampuero and Ben-Zion, 2008; Brietzke et al., 200; Dalguer and Day, 200;

15 Characteristic features of wrinkle-like rupture pulse: ) strong correlation between variations of normal stress and slip 2) strongly asymmetric motion across the fault ) preferred direction of rupture propagation 4) self-sharpening with propagation distance

16 Rupture migration in tri-material structure with multiple available rupture planes (Brietzke and Ben-Zion, 2006) Parameter space study for different Nucleation locations Fault separation Initial shear stress Velocity contrasts

17 nucleation location max. of slip velocity on fault [m/s] distance max. [m] of slip velocity on fault [m/s] distance max. [m] of slip velocity on fault [m/s] distance max. [m] of slip velocity on fault [m/s] Coulomb friction Example results distance max. [m] of slip velocity on fault [m/s] distance max. [m] of slip velocity on fault [m/s] distance max. [m] of slip velocity on fault [m/s] distance max. [m] of slip velocity on fault [m/s] distance max. [m] of slip velocity on fault [m/s] distance [m] nucleation location max. of slip velocity on fault [m/s] distance max. [m] of slip velocity on fault [m/s] distance max. [m] of slip velocity on fault [m/s] distance max. [m] of slip velocity on fault [m/s] Regularized Prakash-Clifton friction distance max. [m] of slip velocity on fault [m/s] distance max. [m] of slip velocity on fault [m/s] distance max. [m] of slip velocity on fault [m/s] distance max. [m] of slip velocity on fault [m/s] distance max. [m] of slip velocity on fault [m/s] distance [m]

18 What happens if we add additional ingredients? Stress heterogeneities (Ben-Zion & Andrews, 8; Andrews & Harris, 200) Low-velocity fault zone layer (Harris & Day, ; Ben-Zion & Huang 2002; Brietzke & Ben-Zion, 2006) Viscosity in the bulk (Harris and Day, ; Brietzke and Ben-Zion, 2006) Prakash-Clifton friction (Cochard & Rice, 2000; Ben-Zion & Huang, 2002) Contrast of permeability structure (Rudnicki & Rice, 06; Dunham & Rice, 08) Slip-weakening friction (Harris & Day, ; Shi & Ben-Zion, 06; Rubin & Ampuero, 200; Brietzke et al., 200, 2008 ) Creation of off-fault damage (Ben-Zion and Shi, 200; Duan, 2008) Multiple possible rupture plans (Brietzke and Ben-Zion, 2006) Velocity-weakening friction (Ampuero and Ben-Zion, 2008) D effects (Brietzke et al., 200, 200; Dalguer & Day, 200) There is a diversity of phenomena. However, the results show collectively that ruptures evolve for broad ranges of realistic conditions to slip pulses in the preferred direction.

19 Effective behavior on different scales: single microcrack, macroscopic shear crack, cataclastic flow, fault zone, fault system, plate boundary,

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21 Frameworks for studying brittle deformation σ σ σ Rock strength Continuum Mechanics Fracture mechanics Friction studies K, G ε σ Damage rheology Granular Mechanics statistical mechanics

22 When material is stressed it deforms Rock strength experiments BC are very important! Faulting angle typically around 0 degrees!

23 Brittle-ductile: stress-strain curves with permanent inelastic brittle and ductile deformation brittle ductile ductile brittle Increasing pressure leads to ductility Increasing temperature leads to ductility

24 Stress-strain and AE locations for Westerly granite (Lockner et al., 2)

25 Lockner s animations Acoustic emission in fracturing experiments with Westerly granite and Berea sandstone

26 Fracture, cracking: Deformation Mechanisms t G c t + Δt x 0 X 0 + Δx General definition: localized deformation converting elastic strain energy to surface energy G c associated with cohesion (analogous to latent heat in solidliquid-gas phase transitions) Energy sinks: Quasi-static tensile Dynamic tensile Dynamic shear Surface area Surface area Kinetic energy (radiation) Plastic strain and damage Surface area Radiation Heat (friction) Plastic strain and damage

27 Friction: σ n τ τ μ = τ/σ n General definition: localized deformation associated with sliding on existing surface (no large-scale extension of surface area, but microscopic contact area decreases). In pure frictional sliding across fault, the released strain energy is converted to heat and seismic radiation.

28 Plastic flow: General definition: distributed deformation of solid associated with internal motion of defects ( dislocations ) in lattice. Surface area is conserved. The released strain energy is converted to heat (and some radiation). Highly enhanced by increasing T, which improves the dislocations mobility, and P.

29 Viscous flow: General definition: distributed deformation associated with fluidlike flow satisfying τ = f (ε& ) E.g., for linear Newtonian viscosity τ ij = D & ε ijkl kl which in D is τ = ηε& Strain energy is converted to heat. Creep is a form of distributed or localized viscous flow associated with dislocations and diffusion of material

30 Cataclastic flow: Mega-breccia Micro-breccia General definition: distributed brittle deformation associated with motion of rock particles on a large collection of cracks and/or frictional surfaces Strain and gravitational energy converted to heat (and in dynamic events also radiation and fracturing)

31 Viscoelasticity: distributed deformation that is elastic on short time scales and viscous on long ones. (a) Maxwell viscoelasticity: σ σ 0 (b) Kelvin-Voigt viscoelasticity: ε 0 ε & ε = σ& + μ t t E σ = σ η v με + η & ε Standard viscoelasticity is (a) in parallel with a spring Since there is no clear criterion on what are small and large time scales, most materials belong to this category. In both Maxwell and KV viscoelasticity, the characteristic time scale for relaxation is T = η/μ. The response of material at different time scales may be characterized by Deborah number D=T/t 0, where T is relaxation time and t 0 is time scale of interest E.g., D >> ~ elastic

32 Brittle failure localized deformation with fracture, friction and increasing surface area Sharp tip and/or surface Stable or unstable (usually unstable) Failure (stress drop) under small strain Strong dilatancy: ΔV/V > 0 ( swelling of the deforming material) Strong dependency on σ n (because of dilatancy and friction) Weak dependency on T (in the brittle range) jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj Favored by low T, P, e.g., z <. km Ductile flow distributed deformation with viscous, plastic, creep (~constant surface area) No sharp tip and/or surface Stable or unstable (usually stable) Ability to sustain large strain without failure no dilatancy kjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj weak dependency on σ n kjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj u/~slombey/asci/solids/fra cture/ strong dependency on T (because of increasing mobility with increasing T) Favored by high T, P e.g. < z km Deformation in the range. < z < km is mixed plastic-brittle or semi-brittle The mode of deformation depends strongly on space-time scales!!!

33 Brittle - Ductile transition: frictional τ s μσ n 0.( ρ ρw ) gz creep & ε A 0 e Q / RT n τ Scholz 2002 This only demonstrates plausibility! Other mechanisms can produce similar distributions. Be aware of non-uniqueness!

34 Seismicity on a model fault governed by friction and creep (Ben-Zion, 6) Can get similar distributions also with rate-state and damage rheology!